“Everything About You” Ugly Kid Joe
My favorite for breakups (yes, I’m that age), along with Soft Cell’s Tainted Love.
This: http://www.youtube.com/watch?v=t4klR5m4i5M
You wanna know what it’s like
Condemned to live with you
It’s some kind of suicide
Some phase that I’m through
I ain’t sadistic, masochistic
You and me are through
I`m sick to death of everything you do
And if I’m gonna have a puke you bet yer life I’ll puke on you
I’m gonna be, gonna be sick on you
I’m gonna be, gonna be sick on you
I’m gonna be, gonna be sick on you
All down your face, your dress, your legs and your shoes…
Sick on you
Give it to me babe
Give it to me babe
Give it to me babe
It’s been six weeks and thats six weeks too long You thought we were Romeo and Juliet You couldn’t have been more wrong So pack your rags in to your bags Get back where you belong Im sure they miss you, it`s really quite a shame
I hear Skid Row’s just never been the same
End of Discussion.
But if you must sing more than one, mine the first 4 Violent Femmes albums for a dozen or more top rate choices.
I’d be hard pressed to call that a breakup song, unless you mean breaking up with the folkies.
Barnes & Barnes has one which probably expresses the real feelings of many a jilted lover - I Hope She Dies.
While I haven’t heard many of the songs posted here, do remember the OP is attempting to coax someone into actually singing a song.
That said, I second The Breakup Song. And not just because it’s one of my faves. It is a very simple song, lyrically. No fancy notes to have to worry about it hitting or anything like that
Gene Pitney, “24 Hours From Tulsa”. ‘Dearest Darling, I had to write to say that I won’t be home any more’…met another girl, swept off feet by her, ends "and I can never ever …go home again’.
Well, Laughing In Your Face by Lillian Axe is the single best “screw you, we’re over, don’t need your crap anymore” song ever written – unfortunately, there isn’t a karaoke place on the planet that would have it.
Good call.
I don’t love you any more
I don’t think I ever did
And if you ever had any kind of love for me
You kept it all so well hid.
Ouch.
Depending on the dumpee, “You’re So Vain” can be a great breakup song.
“When You Go” by Jonathan Coulton. The MP3 is free for download from his site, but the karaoke version you’ll have to buy.
Please come back and let us know which ones worked out. 
*** Ponder
2 by Tom Petty
“Hope You Never”
(“hope you never fall love with, somebody like you”)
“Feel a Whole Lot Better”
("…and I’ll probably feel a whole lot better, when you’re gone")
Well, the girl who lived in the dorm room next to mine my sophomore year seemed to think it was “Just Once” by James Ingram. She played it twenty-three times in a row the evening she got dumped by her boyfriend.
50 ways to Leave Your Lover
“Just Another Picture to Burn” - Taylor Swift
"Hey Now, Little Darlin’ " - Shelby Lynne
“Hit the Road Jack” - Ray Charles
Your Guts (I Hate 'Em) by Reel Big Fish, but only if it’s a really, really bad breakup.
This is your song
Congratulations
You’re the inspiration
I hate you so much
That I wrote this song
Congratulations
You’re immortalized
I hate so much
I hope you fuckin’ die
One More Minute-- Weird Al Yankovic
What’s the twist?
And I would second “Love Stinks” - shout it out of her system.
I really like “Sometime Around Midnight” by The Airborne Toxic Event.
Oh, my, the memory!
Senior year of college (1986), my roommate got dumped by a girl in our (co-ed) dorm. Said girl believed that she and my roommate were still friends…a sentiment that he did not share.
A month or so after the break-up, she asked my roommate if he’d dub a cassette tape for her (copying some album or other). He agreed…and then he and I spent the next few hours creating a tape that was full of “One More Minute”, played over and over and over.
Well, we thought it was funny. 
Take What You Want and No Man’s Mamma by Ethel Waters fit the bill.
A good portion of Fleetwood Mac’s album Rumours is about breaking up - Second Hand News, Dreams, Go Your Own Way (mentioned above), Silver Springs, Don’t Stop, probably some others.